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EXAMPLE
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G.f.: x = 1*G_1(x) + 1*G_2(x)^2 + 4*G_3(x)^3 + 28*G_4(x)^4 + 278*G_5(x)^5 + 3558*G_6(x)^6 + 55600*G_7(x)^7 +...+ a(n)*G_n(x)^n +...
where the initial iterations of G(x) = (x-x^2) begin:
G_1(x) = x - x^2;
G_2(x) = x - 2*x^2 + 2*x^3 - x^4;
G_3(x) = x - 3*x^2 + 6*x^3 - 9*x^4 + 10*x^5 - 8*x^6 + 4*x^7 - x^8;
G_4(x) = x - 4*x^2 + 12*x^3 - 30*x^4 + 64*x^5 - 118*x^6 +-...;
G_5(x) = x - 5*x^2 + 20*x^3 - 70*x^4 + 220*x^5 - 630*x^6 +-...;
...
Coefficients in the n-th power of the n-th iteration of x-x^2 begin:
G_1(x)^1: [1, -1, 0, 0, 0, 0, 0, ...];
G_2(x)^2: [1, -4, 8, -10, 8, -4, 1, 0, 0, 0, 0, 0, ...];
G_3(x)^3: [1, -9, 45, -162, 462, -1095, 2217, -3900, 6024, ...];
G_4(x)^4: [1, -16, 144, -952, 5120, -23608, 96296, -354664, ...];
G_5(x)^5: [1, -25, 350, -3600, 30225, -218775, 1410780, ...];
G_6(x)^6: [1, -36, 720, -10530, 125400, -1286376, 11759727, ...];
...
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PROG
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(PARI) /* n-th Iteration of a function: */
{ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
/* G.f.: */
{a(n)=local(G=x); if(n<=1, G=x, G=x-sum(k=1, n-1, a(k)*ITERATE(k, x-x^2, n)^k)); polcoeff(G, n)}
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